Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol22.Issue8.1999:

Second-order characteristic methods for advection±

di€usion equations and comparison to other

schemes

1

Mohamed Al-Lawatia

a,*

_{, Robert C. Sharpley}

b

_{& Hong Wang}

b a

Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, Al-Khod Postal Code 123, Muscat, Oman

b

Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208, USA

(Received 3 February 1998; revised 27 August 1998; accepted 14 September 1998)

We develop two characteristic methods for the solution of the linear advection di€usion equations which use a second order Runge±Kutta approximation of the characteristics within the framework of the Eulerian±Lagrangian localized adjoint method. These methods naturally incorporate all three types of boundary con-ditions in their formulations, are fully mass conservative, and generate regularly structured systems which are symmetric and positive de®nite for most combina-tions of the boundary condicombina-tions. Extensive numerical experiments are presented which compare the performance of these two Runge±Kutta methods to many other well perceived and widely used methods which include many Galerkin

methods and high resolution methods from ¯uid dynamics. Ó _{1999 Elsevier}

Science Ltd. All rights reserved

Key words:Characteristic methods, Comparison of numerical methods, Eulerian± Lagrangian methods, Numerical solutions of advection±di€usion equations, Runge±Kutta methods.

1 INTRODUCTION

Advection±di€usion equations are an important class of partial di€erential equations that arise in many scienti®c ®elds including ¯uid mechanics, gas dynamics, and at-mospheric modeling. These equations model physical phenomenon characterized by a moving front. In ¯uid dynamics, for example, the movement of a solute in ground water is described by such an equation. Since these equations normally have no closed form analytical solutions, it is very important to have accurate numer-ical approximations. When di€usion dominates the physical process, standard ®nite di€erence methods (FDM) and ®nite element methods (FEM) work well in solving these equations. However, when advection is the

dominant process, these equations present many nu-merical diculties. In fact, standard ®nite element and ®nite di€erence methods produce solutions which ex-hibit non-physical oscillations, excessive numerical dif-fusion which smears out sharp fronts, or a combination of both22;36_.

Many specialized schemes have been developed to overcome the diculties mentioned. One class of these methods, usually referred to as the class of Eulerian

methods, uses an Eulerian ®xed grid and improved

techniques, such asupstream weighting, to generate more accurate approximations. Most of the methods in this class are characterized by ease of formulation and im-plementation, however their solutions su€er from ex-cessive time truncation errors. Moreover, they put a strong limitation on the Courant number and hence require very small time steps to generate stable solu-tions. Among these methods are the Petrov±Galerkin FEM methods1,4,7,10,60which are improvements over the standard Galerkin FEM that incorporate upwindingin Ó1999 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0309-1708/99/$ ± see front matter

PII: S 0 3 0 9 - 1 7 0 8 ( 9 8 ) 0 0 0 3 5 - 9

*_{Corresponding author.} 1

This research was supported in part by DOE Grant No. DE-FG05-95ER25266, by ONR Grant No. N00014-94-1-1163, and by funding from the Mobil Technology Company.

the space of the test functions. Also included in the class of Eulerian methods are the streamline di€usion FEM methods (SDM)5,20,27,32±34,36,37 and the continu-ous and discontinucontinu-ous Galerkin methods (CGM, DGM)24,35,38,46,47. The SDM improve over the standard space-time Galerkin FEM by adding a multiple of the (linearized) hyperbolic operator of the problem consid-ered to the standard test functions. Thus they add nu-merical di€usion only in the direction of the streamlines. The SDM formulations have a free parameter which determines the amount of di€usion applied and there-fore has a great e€ect on the accuracy of these methods. In practice, this parameter should be large enough to avoid oscillations in the solution, but not too large to damp the solution. A clear choice of this parameter is not known, in general, and is heavily problem depen-dent. The CGM and DGM are well suited for purely hyperbolic equations and recently have been extended to solve the advection±di€usion equations. They are space-time explicit methods in which, starting with the initial time solution and Dirichlet data at the in¯ow boundary, one would successively iterate over the elements of a quasi-uniform triangulation of the space-time domain in an order consistent with the domain, solving a local system over each element. In addition to the methods mentioned above, the class of Eulerian methods includes the high resolution methods in ¯uid dynamics such as the Godunov methods, the total variation diminishing methods (TVD), and the essentially non-oscillatory methods (ENO)11±13,19,25,28,51±54. These methods, as well as the CGM and DGM, are well suited for advection-di€usion equations with small advection-di€usion coecients and in general impose an extra stability restriction on the size of the time step taken based on the magnitude of this coecient. Therefore, they are very sensitive to changes in the di€usion coecient, which in practical problems is likely to have large values at certain points.

Another class of methods, usually known as

charac-teristic methods, makes use of the hyperbolic nature of

the governing equation. These methods use a combi-nation of Eulerian ®xed grids to treat the di€usive component, and Lagrangian coordinates by tracking particles along the characteristics to treat the advective component. Included in this class are the Eulerian± Lagrangian methods (ELM), the modi®ed methods of characteristics (MMOC), and the operator splitting methods2,14,16,18,21±23,26,31,40,42±44,55,61. These methods have the desirable advantage of alleviating the restric-tions on the Courant number, thus allowing for large time steps. Furthermore, the Lagrangian treatment in these methods greatly reduces the time truncation errors which are present in Eulerian methods. On the other hand these methods have diculty in conserving mass and treating general boundary conditions. The Euler-ian±Lagrangian localized adjoint methods (ELLAM) were developed as an improved extension of the char-acteristic methods that maintains their advantages but

enhances their performance by conserving mass and treating general boundary conditions naturally in their formulations. The ®rst ELLAM formulations were de-veloped for constant coecient advection±di€usion equations8,30,48. The strong potential that these FEM based formulations and their numerical results have demonstrated have led to the development of additional formulations for variable-coecient advection di€usion equations49,56and for non-linear equations15 as well as ®nite volume formulations9,29. However, because the characteristics for variable-coecient advection±di€u-sion equations cannot be tracked exactly in general, many characteristic and ELLAM methods that have been developed use a backward Euler approximation for these characteristics due to its simplicity and stability. These (backward Euler) schemes are second order accurate in space but only ®rst order accurate in time.

An ELLAM based formulation which uses a second order approximation for the characteristics was devel-oped recently for ®rst-order advection-reaction equa-tions58. This formulation was shown to be of second order in both space and time. Unlike the ®rst-order equations, similar treatment for the advection±di€usion equations is more problematic since one needs to treat the di€usive component and its partial derivatives (with respect to the rectangular and characteristic coordi-nates) carefully along the characteristics to produce systems having desirable structure. In addition, boun-dary treatment is more involved because at both the in¯ow and out¯ow boundary data can be speci®ed in many forms which then need to be incorporated into the formulations. In this paper we develop characteristic methods for the one-dimensional linear advection±dif-fusion equations, based on a second order Runge±Kutta approximation of the characteristics. We present a backward tracking (BRKC) and a forward tracking (FRKC) Runge±Kutta characteristic scheme, both of which are mass conservative and incorporate boundary conditions naturally in their formulations. These meth-ods, which can be thought of as generalizations of the ELLAM schemes, generate tridiagonal (regularly structured in multidimensions) matrices which are symmetric (except at the in¯ow boundary), positive de®nite, and therefore easily and eciently solved. Moreover, we provide the results of fairly extensive numerical experiments which compare the performance of the two methods developed in this paper to many of the methods mentioned above.

the two schemes. In Section 4 we give a brief description of some well studied and widely used methods which are known to give good approximations to the advection di€usion equations. Section 5 contains the results of the numerical experiments that compare the performance of the two schemes developed in Section 3 and the other methods described in Section 4.

2 DEVELOPMENT OF THE METHOD

We consider the one-dimensional linear variable-coe-cient advection±di€usion equation di€usion coecient, respectively.D…x;t†is assumed to be positive throughout the domain. To simplify our pre-sentation, we also assumeVa;t_†andVb;t_†are positive, i.e. we setx_ˆaandx_ˆbto be the in¯ow and out¯ow boundaries, respectively. In many advection-dominated applications,jD…x;t†j<jV…x;t†j, therefore, methods de-vised to solve eqn (1) should handle this case accurately. We consider general boundary conditions with any combination of Dirichlet, Neumann, or Robin condi-tions at the in¯ow and out¯ow boundaries. The function g…t†is used to denote the time dependence to specify the boundary condition at xˆa and h…t† for the out¯ow boundary atxˆb. The Dirichlet, Neumann, and Robin conditions are formulated by the requirements

u…c;t† ˆ/1…t†; t2 …0;TŠ;

2.1 Variational formulation and characteristic curves

The domain of problem (1), is _‰a;bŠ ‰0;TŠ which we partition in space and time as follows:

0_ˆt0_<t1_< space-time test functions that vanish outside

‰a;bŠ …tn_;tn‡1

Š, which enables us to concentrate on one time periodtn_;tn‡1

Š. Our test functions are discontinu-ous in time and allow for di€erent meshes at di€erent time periods. For notational simplicity, we suppress the temporal index on the grid. The variational formulation of eqn (1) on the domainX_{ˆ ‰}a;b_Štn_;tn‡1_{Š, obtained} by multiplying that equation by a test functionw(which we describe in more detail below) and integrating by parts, becomes

The principle of the localized adjoint methods (LAM) requires the test functions to be chosen from the solu-tion space of the homogeneous adjoint equasolu-tion

L_{w ÿwtÿV…x;t†wx‡ …D…x;t†wx†}

xˆ0: …5† However, the solution space of this equation is in®nite dimensional. Thus we need to split this equation to de-termine a ®nite number of test functions. In fact, dif-ferent splittings lead to di€erent approximation schemes (see8,57). Due to the Lagrangian nature of the exact so-lution, a natural splitting is

wt‡V…x;t†wxˆ0;

…D…x;t†wx†x ˆ0:

…6_†

The ELLAM generalizes the framework of the localized adjoint methods by requiring the test functions to satisfy the ®rst equation in (6) exactly or even approximately (so that the last term on the left-hand side of eqn (4) vanishes exactly or approximately). However, the EL-LAM does not require the second equation in (6) to be satis®ed, thus one does not have to choose the test functionsw…x;t†to be of a complicated form in space. In our formulation, we choose the test functions to be piecewise linear in space, as in the linear standard FEM, and to be constant along characteristics which we dis-cuss below.

The characteristic curves of eqn (1) are given by the solutions of initial value problems for the ordinary dif-ferential equation

dy

dt ˆV…y;t†: …7†

We denote the characteristic curve emanating from a given point…x;_{t†witht2 ‰t}n_;tn‡1_{Š, by}

yˆX…h; x;_t†;8_†

where h is the time position parameter along that characteristic. Furthermore, we introduce below some notation that is described by the following relations:

We de®ne~xandx_{as the head (using forward tracking)}

and the foot (using backward tracking), respectively, of the characteristics. The foot of a characteristic with head on the out¯ow boundary is denoted by b_{t† for}

t2 ‰tn_;tn‡1_{Š. We also de®ne ~t…x† and t…x† as the exit}

times of the characteristicsX…h;x;tn_{†andX…h;x;t}n‡1_†at the out¯ow and in¯ow boundaries, respectively (see Fig. 1(b)). The general time increment over the domain may then be written as

Dt…x†:_ˆ~_{t…x† ÿt…x†;}

where we de®ne t…x† ˆtn _{for characteristics with feet} not on the in¯ow boundary, and similarly~_{t…x† ˆt}n‡1_for characteristics with heads not belonging to the out¯ow boundary. By implicitly di€erentiating the fourth rela-tion in (9) for t_{x† with respect to x, and the relation}

X…h;b;t† ˆb‡Rh

t V…X…s;b;t†;s†ds, for characteristics X…h;b;t†withh_{2 ‰}tn_{;tŠoriginating at points…b;t†on the} out¯ow boundary, with respect tot, we get the following equations (for partial derivatives of the characteristics)

Xx…t…x†;x;tn‡1† ˆ ÿV…a;t…x†† dt_x†

dx ; Xt…h;b;t†jhˆtˆ ÿV…b;t†;

…10_†

in respective order. In the following section, we consider approximations of the characteristics, de®ned by second order Runge±Kutta approximations.

2.2 Reference equation

After we have introduced the notion of the character-istics, we are able to ®nd a reference equation by ex-panding certain of the integrals in the variational formulation (4). First we assume that we can exactly track the characteristics, thus the same reference equa-tion is generated if we use either forward or backward tracking algorithms.

2.2.1 Treatment of the source term

We start with the source term (second term on the right-hand side of eqn (4)) which we break as follows,

Z

tn‡1

tn

Zb

a

fw dxdtˆ

Z Z

X1

fw dy ds‡

Z Z

X2

fwdy ds

‡

Z Z

X3

fwdy ds: …11_†

Here the region X1 represents all points which lie on characteristics with feet on the in¯ow boundary, X3 represents all points on characteristics with heads on the out¯ow boundary, and ®nally X2 represents the re-mainder ofX(see Fig. 1(a)). For clarity of presentation, we use the variableyto represent the spatial coordinate of any point inX, and reservexfor points on the spatial mesh ofXat timetn_ortn‡1_{, representing either heads or} feet of characteristics. Similarly we let s denote the temporal variable instead oft, which we reserve for the temporal coordinate of heads or feet of characteristics which lie on either the in¯ow or out¯ow boundary. The general time increment can also be described for pointsx at timetn‡1_{or timet}n_{in respective order by}

Dt…I†_x†and Dt…O†_{x†which are de®ned as follows:}

Dt…I†…x† ˆtn‡1ÿt…x†; x2 ‰a;bŠ;

Dt…O†…x† ˆ~_{t…x† ÿt}n_;

x2 ‰a;bŠ; …

12_†

wheret_x†and~_{t…x†extend tot}n_andtn‡1_{, respectively, if} the characteristics that de®ne them lie inX2. With the change of variableyˆy…x;s† ˆX…s;x;tn‡1_{†for the ®rst} integral on the right-hand side of (11), we obtain

Z Z

X1

f…y;s†w…y;s†dy ds

ˆ

Z

~

a

a

Z

tn‡1

tx†

…fw†…X…s;x;tn‡1†;s†Xx…s;x;tn‡1†dsdx

ˆ

Za~

a

Dt…I†_x†

2 f…x;t n‡1

†wx;tn‡1

†

ÿ

‡…fw†…a;t…x††Xx…t…x†;x;tn‡1†

dx‡Ef;X1…w† Fig. 1.Partition ofX: (a) The three subdomainsXi…iˆ1;2;3†,

ˆ

where in the second equality we used a trapezoidal approximation for the inner integral whose error Ef;X1…w† is given below. In the last equality we moved

the second integral to the in¯ow boundary by the change of variablea_ˆXt_x†;x;tn‡1

†and the ®rst equation in (10). The trapezoidal error term introduced is given by

Ef;X1…w† ˆ

sincewis constant along characteristics. With the same change of variable, y ˆX…s;x;tn‡1_{†, and similar} treat-ment as in (13), we expand the second integral on the right-hand side of equation (11) to get

Z Z In this expression the trapezoidal error term is given by

Ef;X2…w† ˆ The treatment of the source term over X3 is slightly di€erent since for the characteristics, we backtrack from

points originating on the out¯ow boundary. With the change of variable yˆX…s;b;t†, the third term on the right-hand side of (11) becomes

Z Z

where in the last equality we used the second equation in (10) for the ®rst integral and the change of variablex_ˆ X…tn_{;b;t† for the second. The trapezoidal error term}

2.2.2 Treatment of the di€usion term

wX…X…t…x†;x;tn‡1†;t…x††

where in the second equality we used a backward Euler approximation for the integral over a;a~†and a trape-zoidal approximation over~a;b†with error given below, and in the third equality we made the change wx…X…s;

Using a trapezoidal approximation for this di€usive ¯ux term overa;a~†when the in¯ow boundary condition is Dirichlet or Robin condition introduces unknowns on the in¯ow boundary and severely complicates the scheme. Therefore we chose to use only backward Euler approximation on that interval. Neumann in¯ow con-dition on the other hand allows us to use the more ac-curate trapezoidal approximation for the di€usive integral uniformly over the whole spatial domain. This treatment does not lead to any non-symmetric terms in the formulation. Details of this treatment and boundary implementation are discussed in the following section.

Finally we treat remaining di€usion integral overX3. With the change of variable yˆX…s;b;t† this integral where in the second equality, we have used the relation wX…b;t†Xt…t;b;t† ˆwt…b;t†for the ®rst integral. The

er-2.2.3 Assembly of the reference equation

Substituting the integrals expanded in eqns (13), (15), (17), (19) and (21) back into the variational equation (4) yields the following reference equation

‡

ED;Xi†which represents the collective approximation

er-ror andRo…w† ˆ

Rtn‡1

tn

Rb

au…wt‡Vwx†dxdt represents the adjoint term which in our case vanishes by eqn (6).

3 NUMERICAL APPROXIMATION

Since we do not impose a particular form for the ve-locity ®eldV…x;t†, explicitly solving the ordinary di€er-ential equation (7) which de®nes the characteristics is not possible in general and introduces additional di-culty. Therefore in our numerical approximation of the solution of the advection di€usion equation (1), we consider an approximation of the characteristics Xh; x;_{t†which is based on a second order Runge±Kutta} approximation known as Heun's method. In particular, we de®ne the approximate characteristic curve emanat-ing from a point…x;_{t†, witht2 ‰t}n_;tn‡1_{Š, by}

Furthermore, if the Courant number Cr_ˆ maxV†Dt=Dx is at least 2, we partition the out¯ow

whereICin this case is the integer part of Cr. When no subdivision of the out¯ow boundary is introduced (Cr<2), we let IC_ˆ1 so that both cases are treated uniformly. This partition introduces some unknowns on the out¯ow boundary, however it has the advantage of insuring that the schemes we develop are suitable even for large Courant numbers. We de®ne the test functions at timetn‡1_{as hat functions given by} tend these test functions de®ned by (26) and (27) to be constant along the approximate characteristics. The test functionwI is a combination of both, that is,wI…x;tn‡1† is de®ned by (26) for the interval_‰xIÿ1;xIŠ, andwI…b;t†is de®ned by (27) on the interval‰tI‡1;tn‡1Š.

The numerical schemes are based on approximating the exact solutionu, which satis®es the reference equa-tion (23), by a piecewise linear funcequa-tionUwhich satis®es a similar equation with two di€erences: (i) the new equation will be developed using the change of variables resulting from the approximate characteristic tracking given by (24), and (ii) the error and adjoint terms, sim-ilar to E…w†andRo in (23), are not included. Since the terms neglected contribute global errors which are of order ODx†2‡ …Dt†2†, the resulting schemes will be of desired accuracy. In the following two subsections we develop two schemes to solve the advection±di€usion equation (1) based on backward and forward charac-teristic tracking, respectively. Here we emphasize that the test functions de®ned above are constant along the approximate characteristics.

3.1 Backtracking Runge±Kutta characteristic scheme

The ®rst numerical scheme (BRKC) is based on a backward tracking of characteristics. We ®rst let the domainsXibe de®ned in a similar manner as before, but use the approximate characteristicsY…h;x;tn‡1_{†in place} of the true characteristics (see Fig. 1(a)). In this case, our characteristics originate at points …x;tn‡1_{† and are} given by Y…h;x;tn‡1_{† in X}

1 and X2. In X3, the charac-teristics originate at points …b;t† on the out¯ow boun-dary and are given byY…h;b;t†. The notation introduced in Section 2 for exact tracking is also used in our nu-merical scheme formulation, however we modify it for this approximate characteristic tracking. Accordingly, we de®ne ~x satisfying xˆY…tn_{; ~x;t}n‡1_{† and xˆ}

Y…tn_;x;tn‡1

†as the head and the foot respectively, of the Runge±Kutta approximate characteristics. The foot of a characteristic with head on the out¯ow boundary is denoted by b_{t† ˆY…t}n_{;b;t† for t2 ‰t}n_;tn‡1

done for eqn (23). Since, however, the solution of u at time level tn‡1 _{is sensitive to errors arising from the} evaluation of the terms of eqn (23) which involve inte-grals of the trial function U…;tn_{† at the previous time} level, these terms must be treated with care. That these integrals are dicult to evaluate (especially in higher dimensions) is due to the fact that the test functions are de®ned by extending back along the approximate char-acteristics their values from time tn‡1 _{and may be} sub-stantially distorted by this process3,49,59. To indicate the possible complications that may arise, we note that quite complicated geometries may occur even in the case of a constant velocity ®eld. For example, when the support of a test function in three dimensions is traced back in time and intersects a boundary of the spatial domain, a four dimensional space-time region with a complicated geometry results as the support of the space-time test function.

The backtracking scheme uses the approximate characteristicY to change variables in each of the inte-grals where the test functions are evaluated at time level tn

‡. In this way, the integrals are rewritten as integrals at

time tn‡1 _{(or, as the case may be, at the out¯ow} boun-dary) of test functions (i.e. hat functions on the grid) at the current time integrated against trial functions eval-uated at the foot of the characteristics. The term `backtracking' comes from the use of the backward characteristics to determine the trial function values at the previous time. Hence after some rearrangement and combining of terms, the reference equation of the piecewise linear trial functions U using Runge±Kutta backward tracking is given by

Zb where in the third integral on the right-hand side,x_isa

on …a;a~†. This general reference equation holds for all nodes xi;…iˆ0;. . .;I† and ti; …iˆI;. . .;I‡IC†, and simpli®es in speci®c cases. The sti€ness matrix is up-dated by evaluating terms involvingU at time leveltn‡1 and at the out¯ow boundary in eqn (28). Likewise, by evaluating the remaining terms, we can update the right-hand side vector of the generated system.

As we mentioned earlier, one diculty arises in ac-curately computing the terms evaluated at feet of char-acteristics (i.e.xort…x†), since several grid points from the previous time level may occur as backtracked points within a given interval at the current time. To illustrate how to overcome this diculty, we will focus on the ®rst term on the right-hand side of eqn (28) applied with test functionwi. In this case, whenxvaries over the interval

‰xiÿ1;xiŠ(i.e., the left half of the support ofwi…;tn‡1†),x will vary over the interval‰xiÿICÿ3;xiÿICŠ. Therefore, due to the distortion by the characteristic tracking,U…x_;tn_† is not a piecewise linear function, but rather a piecewise smooth function of x2 ‰xiÿ1;xiŠ. This possible loss of smoothness, however, a€ects the accuracy of high order quadrature methods. Therefore, we simply subdivide the interval so thatU…x;tn_{†is smooth on each subinterval} and apply numerical integration to each subinterval separately. To illustrate the idea, we suppose xiÿ1 is in

‰xjÿ1;xj† and xi belongs to ‰xj‡1;xj‡2†, then we would split the integral into three parts along the intervals

on this general problem and how to treat multivariate problems may be found in29,40,57.

Another diculty in incorporating the known values of the trial function in eqn (28) arises from evaluating the expressionUx…x;tn†whenx happens to be one of

the nodal pointsfxig I

iˆ0 at whichUx may have a jump

discontinuity. Since our characteristic tracking is only approximate, we assume that all points within a small tolerance of xi belong to the interval …xi;xi‡1†. This introduces an error which is controlled to within the desired order provided the tolerance is of the same order.

The scheme for nodes near the in¯ow boundary (i.e., xi…iˆ0;. . .;IC‡1†) di€ers since the characteristics traced back strike that boundary and lead to boundary terms appearing in the scheme. In the case of Robin ¯ux boundary condition, we can easily change the ®fth in-tegral on the left-hand side of eqn (28) as follows

Z Dirichlet in¯ow boundary condition introduces extra diculty, since then the unknown di€usive ¯ux term in the ®fth integral on the left-hand side of eqn (28) ap-pears in the scheme equations for nodesxi<a~. An al-ternative treatment for the di€usion term discussed in48,57 avoids this diculty. Instead of integrating the term Rtn‡1

tn

Rb

aÿ…Dux†xw dxdt by parts as we did in the derivation of the variational formulation (4) and then applying the integral approximation (trapezoidal, Eu-ler), we can reverse the order, whence we get the term

Za~

instead of the di€usive ¯ux in the ®fth term on the left-hand side of eqn (28). The ®fth term on the left-left-hand side of eqn (28) then simpli®es to

Z

tn‡1

tn

V…a;t†g1…t†w…a;t†dt: …31†

Since the value of the trial functionU is known at node x0 from the prescribed Dirichlet boundary condition, equations need not be formulated there. Hence our scheme will be stipulated only for nodesxi(iˆ1;. . .;I†. However, we do modify w1_ˆw0_‡w1 so that the test functions sum to one in order to maintain mass con-servation.

Neumann in¯ow boundary condition generates a more natural scheme in the sense that a trapezoidal approximation, instead of the backward Euler approximation described above, can be implemented for the di€usive integral over X1. The advantage of this treatment is that it symmetrizes the in¯ow terms thus

maintaining the symmetry of the whole formulation. Implementing this boundary condition leads to the fol-lowing two changes to eqn (28): (i) a factor of 1/2 multiplies the second integral on the left-hand side, and (ii) the additional term of

ÿ

is added on the hand side. The ®fth term on the left-hand side of eqn (28) is replaced by

Z

Wang, Ewing, and Russell57 describe in detail several possible ways to treat the ®rst integral in eqn (33). Their ®ndings suggest combining this term with the ®rst term on the left-hand side of the reference equation (28) and replacing these with the term

Zb

~

a

U…x;tn‡1_†w…x;tn‡1_{†dx: …34†}

Although this term is not exact for a variable velocity ®eldV, the error introduced is within the desired order. Due to the fact that we generate unknowns at the out¯ow boundary at nodesti, boundary treatment here becomes very important. First we note that U…b;tn_{† is} known from the previous time step solution, hence, we do not impose an equation attI‡ICˆtn. Therefore, as we did earlier in the case of Dirichlet in¯ow conditions, we rede®ne the test function wI‡ICÿ1:ˆwI‡ICÿ1‡wI‡IC, to

maintain mass balance. For test functions wi; …iˆI‡1;. . .;I‡ICÿ1†, the terms on the in¯ow boundary and terms evaluated at time tn‡1 _{of eqn (28)} vanish. Thus for nodesti…iˆI‡1;. . .;I‡ICÿ1†, the reference eqn (28) simpli®es to the following:

equation for node xI, has in addition to the terms in eqn (35) some other terms that are in eqn (28) which do not vanish for this node. Neumann out¯ow boundary condition can be incorporated in eqn (35), simply by making the changeÿ…DUx†…b;t† ˆh2…t†in the ®rst and second integrals on the left-hand side of this equation. Similarly the ¯ux out¯ow condition can be incorporated in eqn (35) by applying the condition to the ®rst integral on the left hand side, and using the relation-DUx†

…b;t† ˆh3…t† ÿ …VU†…b;t† for the second integral on the left hand side of eqn (35). In the case of out¯ow Dirichlet condition, the solution U…b;tn‡1

† is known from the boundary condition, therefore the equations for i6Iÿ1 decouple and can be solved for Uxi;tn‡1

†; …i_ˆ0. . .I_ÿ1_†. Unless we desire the values of the derivative of the solution at the out¯ow boun-dary, we do not generate equations there.

3.2 Forward tracking scheme

In this brief section we indicate the development of a scheme (FRKC) based on a `forward tracking' which is a feasible alternative for the backward tracking scheme of the last subsection especially for multidimensional problems. With the test functions as de®ned earlier by eqns (26) and (27), we use the reference equation (23) to derive the numerical scheme for Runge±Kutta forward tracking. Here again we wish to avoid the expensive task of backtracking the geometry to perform the integration of the test functions at the previous time level. Instead we perform this integration by using the ®xed spatial grid of the trial function at tˆtn _{and apply numerical} quadrature. The values of the test function required by the quadrature are obtained by forward tracking the quadrature points and evaluating wi…~x;tn‡1† or its de-rivatives at time tn‡1_{. This forward scheme is only used} to evaluate certain terms to incorporate known values and therefore does not lead to distorted grids that comprise a major drawback of explicit numerical schemes.

In this scheme, the characteristics originate at points

…x;tn_{† and are given byY…h;x;t}n_{†in X}

2 andX3 or they originate at pointsa;t†and are given byY…h;a;t†inX1. Here again we use the same notation established in Sections 2 and 3, except we rede®ne them for this par-ticular forward tracked approximate characteristics. Accordingly, we de®ne~xˆY…tn‡1_;x;tn_{†andxsatisfying}

xˆY…tn‡1_;x;tn_{† as the head and foot of the} approxi-mate characteristic. Moreover, we let t_{x† (given by}

xˆY…tn‡1_{;a;t…x††) and}~_{t…x†(given bybˆY}~_t…x†;x;tn_†) denote the exit time of the approximate characteristics at the in¯ow and out¯ow boundaries, respectively. The time increments _Dt…I† _{and Dt}…O† _{are then given by}

eqn (12), wheret…x†and~_t…x†extend totn _{and t}n‡1_, re-spectively, if they are de®ned by characteristics in X2. The reference equation for the forward scheme, derived in a similar manner as eqn (23), is given by

Zb

The boundary treatment is similar to that of the back-tracking scheme BRKC.

3.3 Experimental order of convergence

max nˆ0;::;NkU…x;t

n

† ÿux;tn

†kLp_a;b†6Ca…Dx†

a

‡CbDt_†b;

pˆ1;2; …37_†

wherea,bgive the order of convergence of the error in space and time respectively, and Ca, Cb are positive constants. To obtain the order of convergence in space we ®x Dtˆ1=500 to insure time truncation errors are

appropriately small. We then perform runs varying Dx and apply a linear regression on theLp …pˆ1;2†norms of the error to determine the parametera. Tables 1 and 3 present the results of these runs and the computed value of the parameter a for the two velocity ®elds de-scribed above. In a similar manner we ®x Dxˆ1=700 and perform runs varyingDtto estimate the value of the

Table 1. Order of convergence in space withVx_;t_{† ˆ}1_‡0:1x

Method Dt Dx L2 Error L1Error

BRKC 1/500 1/40 1:70830110ÿ2 _1.008468

10ÿ2

1/500 1/50 9:76281610ÿ3 _5.012535

10ÿ3

1/500 1/60 6:02882210ÿ3 _2.82296310ÿ3

1/500 1/70 3:98970910ÿ3 _1.73883210ÿ3

1/500 1/80 2:79323410ÿ3 _1.19184010ÿ3

aˆ2:6181 aˆ3:0979

Caˆ270:4686 Caˆ919:3184

FRKC 1/500 1/40 3:28064410ÿ2 _1.56033010ÿ2

1/500 1/50 2:99314910ÿ2 _1.33551710ÿ2

1/500 1/60 2:29038510ÿ2 _1.00502610ÿ2

1/500 1/70 1:41509910ÿ2 _6.05296710ÿ3

1/500 1/80 5:04075710ÿ3 _2.07154010ÿ3

a_ˆ2:4812 a_ˆ2:6757

Caˆ418:3782 Caˆ405:2484

BE-ELLAM 1/500 1/40 1:72784210ÿ2 _1.01976310ÿ2

1/500 1/50 1:00438310ÿ2 _5.125955

10ÿ3

1/500 1/60 6:37085910ÿ3 _2.974945

10ÿ3

1/500 1/70 4:37551310ÿ3 _1.910709

10ÿ3

1/500 1/80 3:21240210ÿ3 _1.356822

10ÿ3

aˆ2:4374 aˆ2:9236

Caˆ138:478 Caˆ481:5463

Table 2. Order of convergence in time withVx;t_{† ˆ}1‡0:1x

Method Dt Dx L2Error L1 Error

BRKC 1/6 1/700 5:65333810ÿ3 _2:07966710ÿ3

1/8 1/700 3:18992010ÿ3 _1:22568010ÿ3

1/10 1/700 2:06691110ÿ3 _8:26831310ÿ4

1/12 1/700 1:46044010ÿ3 _5:99989410ÿ4

1/14 1/700 1:09471610ÿ3 _4:56201410ÿ4

bˆ1:9385 bˆ1:7867

Cbˆ0:1808 Cbˆ0:0508

FRKC 1/6 1/700 5:92033510ÿ3 _2:50317810ÿ3

1/8 1/700 3:31085710ÿ3 _1:40072610ÿ3

1/10 1/700 2:11740510ÿ3 _8:98579710ÿ4

1/12 1/700 1:46717110ÿ3 _6:13858310ÿ4

1/14 1/700 1:07516610ÿ3 _4:520550

10ÿ4

b_ˆ2:0123 b_ˆ2:0222

Cbˆ0:2177 Cbˆ0:0939

BE-ELLAM 1/6 1/700 6:47612210ÿ2 _2:88335710ÿ2

1/8 1/700 4:81777710ÿ2 _2:15136910ÿ2

1/10 1/700 3:83662710ÿ2 _1:71682310ÿ2

1/12 1/700 3:18794110ÿ2 _1:42878610ÿ2

1/14 1/700 2:72712010ÿ2 _1:22366410ÿ2

bˆ1:0207 bˆ1:0115

parameter b. Tables 2 and 4 give corresponding results for b for the two velocity ®elds. From these results we clearly see that the two schemes developed are corrob-orated to be second order in space and time. We also see that BE-ELLAM, as was expected, is second order in space but only ®rst order in time. The orders are more apparent in the more rapidly changing velocity ®eld V ˆ1ÿ0:5x (Tables 3 and 4). Tables 1 and 3

show that when the time step Dt is very small, the BRKC, FRKC, and BE-ELLAM schemes generate so-lutions with comparable errors. This is expected since all the schemes are second-order accurate in space and the temporal errors are negligible. In practice one wishes to use as large as possible time step in numerical simu-lations to enhance the eciency without sacri®cing accuracy. Therefore, Tables 2 and 4 bear more

compu-Table 3. Order of convergence in space withVx_;t_{† ˆ}1_ÿ0:5x

Method Dt Dx L2 Error L1Error

BRKC 1/500 1/60 6:11732310ÿ3 _2.475200

10ÿ3

1/500 1/70 3:49516610ÿ3 _1.323890

10ÿ3

1/500 1/80 2:42972810ÿ3 _9.21363410ÿ4

1/500 1/90 1:84787810ÿ3 _6.92362410ÿ4

1/500 1/100 1:40821310ÿ3 _5.17297810ÿ4

aˆ2:8265 aˆ2:9903

Caˆ610:3031 Caˆ475:0919

FRKC 1/500 1/60 1:15978410ÿ2 _5.08529310ÿ3

1/500 1/70 8:45810110ÿ3 _3.55966510ÿ3

1/500 1/80 5:90256610ÿ3 _2.52829810ÿ3

1/500 1/90 3:96778010ÿ3 _1.71344410ÿ3

1/500 1/100 3:48835310ÿ3 _1.51182710ÿ3

a_ˆ2:4833 a_ˆ2:4856

Caˆ308:5422 Caˆ134:2145

BE-ELLAM 1/500 1/60 6:67337210ÿ3 _2.73318910ÿ3

1/500 1/70 4:12420310ÿ3 _1.573640

10ÿ3

1/500 1/80 3:12445410ÿ3 _1.199037

10ÿ3

1/500 1/90 2:60632810ÿ3 _9.991517

10ÿ4

1/500 1/100 2:24082210ÿ3 _8.839847

10ÿ4

aˆ2:1018 aˆ2:1665

Caˆ33:5250 Caˆ17:3620

Table 4. Order of convergence in time withVx;t_{† ˆ}1ÿ0:5x

Method Dt Dx L2 Error L1Error

BRKC 1/6 1/700 3:97164210ÿ2 _1:45725910ÿ2

1/8 1/700 2:15790010ÿ2 _8:29013210ÿ3

1/10 1/700 1:38735910ÿ2 _5:44808510ÿ3

1/12 1/700 9:79620910ÿ3 _3:88844310ÿ3

1/14 1/700 7:34734610ÿ3 _2:93648410ÿ3

b_ˆ1:9897 b_ˆ1:8896

Cbˆ1:3770 Cbˆ0:4261

FRKC 1/6 1/700 3:86681210ÿ2 _1:33669710ÿ2

1/8 1/700 2:04458510ÿ2 _7:22453510ÿ3

1/10 1/700 1:27662710ÿ2 _4:56453710ÿ3

1/12 1/700 8:76441710ÿ3 _3:148085

10ÿ3

1/14 1/700 6:40156310ÿ3 _2:313990

10ÿ3

b_ˆ2:1208 b_ˆ2:0692

Cbˆ1:7050 Cbˆ0:5394

BE-ELLAM 1/6 1/700 1:26777910ÿ1 _5:01752510ÿ2

1/8 1/700 9:37758610ÿ2 _3:74949510ÿ2

1/10 1/700 7:46579210ÿ2 _3:00481610ÿ2

1/12 1/700 6:21089210ÿ2 _2:51215910ÿ2

1/14 1/700 5:32096810ÿ2 _2:16028410ÿ2

bˆ1:0242 bˆ0:9941

tational relevance since very large time steps are nor-mally desired. One sees that the BRKC and FRKC schemes further reduce the temporal errors in BE-EL-LAM, which are themselves signi®cantly smaller than most Eulerian methods. This justi®es the appropriate-ness of these schemes when large time steps are to be taken. The constantCb is much smaller thanCa for all three schemes corroborating that time truncation errors are smaller than spatial errors, a strong advantage of characteristic methods in general.

4 DESCRIPTION OF SOME OTHER METHODS

The necessity to numerically solve advection-dominated advection±di€usion equation with high accuracy has lead to the development of many specialized methods. In this section we brie¯y describe some well perceived methods which are widely used in practice. In the next section we carry out experiments to compare the performance of the BRKC and FRKC schemes with these methods. In our description of these methods, we impose the same general assumptions on the velocity ®eldV…x;t†and the di€usion coecientD…x;t†as in Section 2 and describe the meth-ods with Dirichlet boundary conditions.

4.1 Galerkin and Petrov±Galerkin ®nite element methods

The linear Galerkin (GAL), quadratic Petrov±Galerkin (QPG), and cubic Petrov±Galerkin (CPG) FEM meth-ods with a Crank±Nicholson time discretization and Dirichlet in¯ow and out¯ow boundary conditions for eqn (1) are described by the following formulation

Zb the intervals de®ned by the partition. The three methods di€er in their choices of the test functions wi…x† …iˆ1;2;. . .Iÿ1†. In the GAL method, the test functions are chosen from the space of the trial func-tions, thus, wi…x† are the hat functions de®ned by eqn (26) where wi…x† replaces wi…x;tn‡1†. The Petrov± Galerkin methods were designed to improve the GAL method by introducing some upwinding in the test space. The QPG methods use test functions, which are the sum of the standard hat functions and quadratic asymmetric perturbation terms, de®ned by1,10

wi…x† ˆ the velocity ®eld and the di€usion coecient over the intervalxiÿ1;xi†. The CPG methods on the other hand use test functions which have symmetric cubic pertur-bation terms added to the hat functions as follows4,60

wi…x† ˆ many other Petrov±Galerkin formulations are also known to solve eqn (1) reasonably well, the coecients used in the QPG and CPG methods here have been chosen optimally6 and these methods are among the most popular ones in practice.

4.2 The streamline di€usion FEM methods

The Streamline Di€usion method (SDM) is applied to the non-conservative form of eqn (1) given by

L_{uut‡V…x;t†ux‡Vx…x;t†uÿ …D…x;t†ux†} x

ˆf…x;t†: …41_†

Here we describe the linear SDM formulation for this equation. We divide the domain into the time slabs

‰a;bŠ …tn_;tn‡1_{Š, and successively on each slab, we seek a} continuous and piecewise linear function U…x;t† (dis-continuous in time at tn _{and t}n‡1_{) which satis®es the}

and tn‡1 _{and is zero at x}

ˆa and x_ˆb. In eqn (42) wn

ˆlimt!…tn_†w…x;t†, U_ÿ0 ˆuo…x†, and d is a free

pa-rameter, described below, that has signi®cant in¯uence on the accuracy of the scheme. There are many relations that can be used for the parameter d. One of the most widely used relation is dˆC…h= 

1‡V2

p

†, where h is the mesh diameter and C is a constant to be chosen36. The choice of C determines the amount of di€usion applied in the direction of the characteristics and therefore has a great e€ect on the accuracy of the scheme. One requires thatCis large enough to produce non-oscillatory solution, but not too large to damp the solution. This choice is, in general, problem dependent and not clear in practice. Although there are more im-proved versions of SDM method with shock capturing capacity which produce better approximations, they usually have non-linear formulations (even though they model linear equations), have more free parameters similar to d (described above) that need to be chosen carefully, and also have higher computational cost. The formulation (42) is the one we chose for numerical ex-periments described in the next section.

4.3 The continuous and discontinuous Galerkin FEM methods

The Continuous and Discontinuous Galerkin methods (CGM and DGM)46,47 apply to the non-conservative form of eqn (1) given by equation eqn (41). The domain

Xgˆ …a;b† …0;T†is divided into a quasi-uniform tri-angulation with side length h, and Dirichlet data is as-sumed on the in¯ow portion of the boundary denoted by

CI† and given by V…x;t† n<0, where V…x;t† ˆ

…V…x;t†;1_† gives the characteristic direction and n_ˆ …n1;n2_†is the outward unit normal vector. The boundary data given at the out¯ow portion of the boundary are not used in the formulation. In the DGM method35,38,47 one seeks a discontinuous approximationU…x;t†, which lies in the spacePn…T†of polynomials of degree at most n on each triangleT_{, and satis®es the equation}

Z any sides on the boundary of the domain,

U_{x;t† lim}

e!0‡U……x;t† eV†, andUÿ is the solution

at the previous element or an interpolation of the pre-scribed Dirichlet data for sides onCI†. The CGM24,46is

formulated in the same way by requiring a solution U…x;t† 2Pn…T† which satis®es eqn (43), however there are two main di€erences: First, the trial functionU…x;t†

is required to be continuous over the domainXg, and the test functions are in Pnÿq…T_†…T†, where q…T† is the number of in¯ow sides thatT _{has. Secondly, the} con-tinuity requirement in CGM makes the second terms on both sides of eqn (43) cancel each other. In both of these methods, one iterates over the elements solving a linear system of order equal to the degree of freedom on each element which makes these schemes quasi-explicit. In the next section, we consider the lowest-order CGM (n_ˆ2), and DGM (n_ˆ1).

4.4 High resolution methods (MUSCL and MI N M O DI N M O D)

High resolution methods from computational ¯uid dy-namics are known to be good for purely hyperbolic equations. An extension of these methods to eqn (1) is based on time-splitting of the equation, as described below, and then a high-order Godunov method can be used to solve the advective part, with a mixed FEM method to solve the di€usive part17. We consider two such schemes, the ®rst based on Monotone Upstream-centered Scheme for Conservation Laws (MUSCL) which was developed by van Leer54, and a second, based on a generalization of the ®rst, called the MI N M O DI N M O D,

which was developed by Harten et al.28,51. Assuming thatUn_{x†approximates the solutionu…x;t}n_{†of eqn (1),} we can generate an approximation of ux;tn‡1

† as fol-lows: First the MUSCL or MI N M O DI N M O D scheme, described

below, can be applied to ®nd a solution of the advective equation

†. We note the well-known fact that the mixed method in lowest-order approximation space and a trapezoidal rule of integration is equivalent to the block-centered ®nite di€erence scheme22_.

Now we describe the MUSCL and MI N M O DI N M O D

schemes. Unlike the other methods discussed here which are node based, MUSCL and MI N M O DI N M O Dare cell-centered

assume that the partition (2) is uniform. The MUSCL or

where the Courant number is assumed not to exceed one (a stability requirement) and the left state Un

iÿ1=2;L is

The two methods di€er in the choice of the slopedUn iÿ1=2.

where the di€erence operators are given by

DÿUinÿ1=2ˆ

The MUSCL formulation on the other hand uses the following de®nition for the slope

dUinÿ1=2 1 otherwise, which is the upper bound that allows the steeper representation of sharp fronts.

5 NUMERICAL EXPERIMENTS

In this section we describe numerical experiments which we use to compare the Runge±Kutta characteristic methods developed in this paper (using both forward-and back-tracking) with several generally well-regarded numerical schemes, including various Galerkin and Petrov±Galerkin ®nite element methods, streamline di€usion ®nite element methods, continuous and dis-continuous Galerkin methods, and high resolution methods in computational ¯uid dynamics. We apply these methods to two standard test problems (a smooth Gaussian distribution and a step function) for which we have analytic solutions to the advection±di€usion equation. In addition, each of these functions are typi-cally used to test for numerical artifacts of proposed schemes, such as numerical stability and di€usion, spu-rious oscillations, phase errors, and Gibbs type e€ects near sharp fronts.

In order to test these proposed schemes for advec-tion-dominated transport, we consider the model equa-tion (1) over the time period t2 ‰0;1Š with a variable velocity V…x;t† ˆ1_‡0:1x for Test Problem I and a constant velocity ®eld ofV…x;t† ˆ1 for Test Problem II, and a relatively small di€usion coecient of Dˆ10ÿ4_. We consider both test problems for our initial-boundary conditions with the results for the Gaussian shown in Figs. 2±8 beginning with label I and the step function in Figs. 9±13. For clarity of exposition, we have arranged the numerical methods into 5 groups based on common characteristics of their behavior and implementation. These groups are organized according to the following table:

In our experiments, we have systematically varied the space and time steps to examine the performance of each method. For each grouping we have chosen to display 3 plots which provide a fair illustration of the accuracy of each method, their potential bene®cial properties, as well as their possible undesirable nu-merical artifacts. For comparison to BRKC and FRKC, the ®rst plot in each ®gure, labeled (a), pre-sents the evolved solution for all methods in the group Group Methods

1 Runge±Kutta characteristic methods (BRKC and FRKC)

2 Crank±Nicholson FEM (Galerkin, Quadratic and Cubic Petrov±Galerkin)

3 Streamline Di€usion (with various selections of the control parameter)

4 Continuous and Discontinuous Galerkin 5 High resolution methods in ¯uid dynamics

using a common space mesh (Dx_ˆ1=60 for Problem I,

Dxˆ1=100 for Problem II) and a time step as close as possible to the BRKC/FRKC time step of Dtˆ1=10, but chosen small enough to ensure stability. The third plot of the ®gure (labeled (c)) for each grouping shows the solutions with an optimally ecient and reasonable choice of space and time steps to produce a qualita-tively comparable solution to that of the BRKC and FRKC schemes in Figs. 2 and 9, respectively. The second plot in each ®gure (labeled (b)) shows an in-termediate stage for each grouping. For example, Fig. 5 consists of 3 plots ((a)±(c)) which shows the solutions for model problem I at time T ˆ1 for the methods in group 4 (continuous and discontinuous Galerkin ®nite element methods) withDx;Dt_†taken as

…1=60;1=60†, 1=120;1=180†, and 1=180;1=180†, re-spectively. The only exception to this labeling pattern is the collection of plots for the TVD schemes (MUSCL and MI N M O DI N M O D) applied to model problem I,

where we have included additional plots (Figs. 7 and 8) of solutions to simpli®ed problems to help explain what appears at ®rst to be atypical behavior for these schemes. More explanation is provided at the end of Section 5.1.

To gauge algorithm eciency, we also compare the timings for each method with di€erent space and time steps, especially the timings for each method to achieve the accuracy depicted in plot (c) of each ®gure. These results are presented in Tables 5±9, using the groupings of algorithms as we described in our list above. We have presented bothL2_andL1_{errors in the tables, since theL}2 error is often used for linear advection±di€usion equa-tions while the L1 _{error is preferred for ®rst-order} hy-perbolic equations. We realize, of course, that some code optimization may be possible but feel that these timings are representative of each scheme's eciency on these model problems.

5.1 Model problem I: Gaussian

Our ®rst model problem considers the transport of the one dimensional Gaussian hill over the spatial domain

‰0;1:4Š. The initial condition is given by

u0…x† ˆ exp _ÿ…xÿx0† 2

2r2

!

; …54_†

where the center x0_ˆ0 and the spread r_ˆ

10ÿ3=2_{0:0316, which accordingly determines the} steepness of the Gaussian. In case the coecientsV and Dare constant, it is easy to verify that

ux;t_{† ˆ}pjt_†exp _ÿ1

2j…t†g…x;t† 2

…55_†

is the exact solution to the homogeneous equation (1) (i.e.,f…x;t† ˆ0) where

g…x;t† ˆxÿx0ÿVt

r ; j…t† ˆ 1‡

2Dt

r2

ÿ1

: …56_†

We have chosen to display the results for relatively small di€usion (Dˆ10ÿ4_{) and a relatively simple linear} ve-locity ®eldV…x† ˆ1‡ax, since these are representative of results we have observed for both constant and variable coecient equations. In this case the model equation (1) is no longer homogeneous, and the right-hand side becomes

f…x;t†

ˆ ÿ a

r4j…t† 5=2

P…x;t;a;D;r† exp

ÿ1

2j…t†g…x;t† 2

…57_† whereP is a multivariate polynomial de®ned by

P…x;t;a;D;r_{† ˆ}aDV…x†2t4ÿ2aD D… ‡ …xÿx0†V…x††t3

‡ 2D…xÿx0† ‡aD 3x2ÿ4x0x‡x20

ÿ

ÿr2

‡r2

V…x†2t2

ÿ 2Dx_ÿx0_†2_‡r2

‡r2

…x

ÿx0_†Vx_†t_ÿr4_:

…58_† As in the earlier experiment in Section 3.3 for deter-mining the order of convergence, we setaˆ0:1.

Notice that in practical simulations, a ®xed (relatively coarse) spatial mesh is often given and a time step is often chosen as large as possible to enhance the e-ciency of simulations without sacri®cing accuracy. Hence, in our numerical experiments we have picked baseline parameters of _Dxˆ1=60 and Dtˆ1=10, for each numerical scheme in our testbed. We choose the spatial grid size ofDxˆ1=60 so that the steep analytical solution can be represented accurately, with which we compare all the numerical solutions. With the relatively coarse time step ofDtˆ1=10, the Runge±Kutta and the ELLAM schemes have already generated accurate

nu-Table 5. Results for BRKC & FRKC. (CPU is in seconds)

Method Dt Dx L2Error L1Error CPU Figures

BRKC 1/5 1/60 8:37097310ÿ03 _3:15161310ÿ3 _{1.2 ±}

1/10 1/60 2:58154710ÿ03 _9:34895510ÿ4 _{2.1 2}

FRKC 1/5 1/60 8:42696010ÿ03 _3:57126210ÿ3 _{0.7 ±}

merical solutions. However, all other methods in the testbed have implicit Courant restrictions on the time step either for the reason of stability or for the reason of accuracy, hence the time step may not be chosen as large as that permitted by the Runge±Kutta and ELLAM schemes. Therefore, for each of these methods (with the exception of MUSCL and MI N M O DI N M O D schemes), we ®rst

®x the space grid size of_Dxˆ1=60 and choose the time steps of_Dtˆ1=60, 1=120, 1=180, and 1=500. The time step of _Dtˆ1=60 is the largest possible time step for these methods to generate reasonable numerical solu-tions, while the time step of Dtˆ1=500 is chosen to observe the performance of these methods for very ®ne time steps since in advection-dominated transport the temporal errors dominate the numerical solutions. We then reduce the space grid size Dx to Dx_ˆ1=120 and 1=180 and use the same time steps above to observe the

e€ect of spatial errors. In this way, we can see what combination of DxandDt (and so the CPU) used with each of these methods can generate solutions compara-ble to the two Runge±Kutta methods (BRKC and FRKC) withDxˆ1=60,Dtˆ1=10. As for the MUSCL and MI N M O DI N M O D schemes, since the maximum velocity is

1.14 over the interval [0,1.4], we were required by the CFL constraint to begin with_Dtˆ1=69 forDxˆ1=60, with _Dtˆ1=137 for Dxˆ1=120, and with Dtˆ1=206 for _Dxˆ1=180. The smallest time step is Dtˆ1=700. We also used ®ner space grid sizes of _Dxˆ1=300 and 1=400 to match qualitatively the results of other schemes.

Fig. 2 shows the initial condition for model problem I (plotted with solid line at the in¯ow boundary) along with the evolved solution at time t_ˆ1. Near the right boundary, we see the analytic solution (solid line), the

Table 6. Results for GAL, QPG, and CPG (CPU time is in seconds)

Method Dt Dx L2 Error L1Error CPU Figures

GAL 1/60 1/60 7:47778910ÿ2 _4:676085

10ÿ2 _{15.6 ±}

1/69 1/60 6:31593110ÿ2 _3:769911

10ÿ2 _{17.9 3(a)}

1/120 1/60 2:92831710ÿ2 _1:51394010ÿ2 _{31.2 ±}

1/180 1/60 1:60641610ÿ2 _7:87631710ÿ3 _{46.3 3(b)}

1/500 1/60 5:74297510ÿ3 _2:79745410ÿ3 _{128.9 ±}

1/60 1/120 7:48448510ÿ2 _4:56036910ÿ2 _{31.2 ±}

1/120 1/120 2:81034910ÿ2 _1:33249510ÿ2 _{61.9 ±}

1/180 1/120 1:39430210ÿ2 _6:16647810ÿ3 _{92.7 ±}

1/500 1/120 2:00613110ÿ3 _8:10081810ÿ4 _{257.0 ±}

1/60 1/180 7:49426110ÿ2 _4:56615810ÿ2 _{46.3 ±}

1/120 1/180 2:81624810ÿ2 _1:33293310ÿ2 _{92.7 ±}

1/180 1/180 1:39576710ÿ2 _6:17422510ÿ3 _{138.6 ±}

1/500 1/180 1:91003310ÿ3 _8:03627410ÿ4 _{385.5 3(c)}

QPG 1/60 1/60 6:16516710ÿ2 _3:29949910ÿ2 _{15.6 ±}

1/69 1/60 5:04262610ÿ2 _2:59711310ÿ2 _{17.9 3(a)}

1/120 1/60 2:44446010ÿ2 _1:14713010ÿ2 _{31.2 ±}

1/180 1/60 2:32904610ÿ2 _1:09544710ÿ2 _{46.3 3(b)}

1/500 1/60 2:74578510ÿ2 _1:292518

10ÿ2 _{128.9 ±}

1/60 1/120 7:18244010ÿ2 _4:207188

10ÿ2 _{31.2 ±}

1/120 1/120 2:49763610ÿ2 _1:171133

10ÿ2 _{62.0 ±}

1/180 1/120 1:12081210ÿ2 _5:02541110ÿ3 _{92.7 ±}

1/500 1/120 4:47246310ÿ3 _1:90725010ÿ3 _{257.5 ±}

1/60 1/180 7:38214510ÿ2 _4:43349910ÿ2 _{46.3 ±}

1/120 1/180 2:67619510ÿ2 _1:25720710ÿ2 _{92.7 ±}

1/180 1/180 1:24890610ÿ2 _5:54715610ÿ3 _{138.7 ±}

1/500 1/180 1:28615010ÿ3 _5:29896510ÿ4 _{385.5 3(c)}

CPG 1/60 1/60 Unbounded 15.6 ±

1/69 1/60 1:50276210ÿ2 _6:29921910ÿ3 _{18.0 3(a)}

1/120 1/60 6:80306910ÿ3 _3:01001810ÿ3 _{31.3 ±}

1/180 1/60 5:12537110ÿ3 _2:40098510ÿ3 _{46.6 3(b)}

1/500 1/60 4:47692410ÿ3 _2:21843310ÿ3 _{129.6 ±}

1/60 1/120 Unbounded 31.3 ±

1/120 1/120 Unbounded 62.3 ±

1/180 1/120 2:22188510ÿ3 _8:849107

10ÿ4 _{93.2 ±}

1/500 1/120 5:70042710ÿ4 _2:392513

10ÿ4 _{258.9 ±}

1/60 1/180 Unbounded 46.6 ±

1/120 1/180 Unbounded 93.2 ±

1/180 1/180 Unbounded 139.8 ±

back-tracked BRKC solution (marker o), and the for-ward-tracked FRKC solution (dotted line). Both meth-ods use a spatial step ofDxˆ1=60 with a relatively large time step of Dtˆ1=10. Each of the two solutions give very accurate approximations which are free of numer-ical oscillations, arti®cial di€usion, phase error, and adverse boundary e€ects. The timings for the BRKC and FRKC schemes are presented in Table 5 and pro-vide baseline timings for all experiments.

Fig. 3(a) contains the plots of the analytic solution (solid line), the Galerkin approximation (labeled with symbol +), quadratic Petrov±Galerkin (dotted line) and cubic Petrov±Galerkin (labeled with symbol o) at time tˆ1 with Dxˆ1=60 where the time-stepping method employed is Crank-Nicholson. Initially, a time step of

Dt_ˆ1=60 was to have been used, however the CPG methods generated unbounded solutions. Hence in Fig. 3(a), a time step of Dtˆ1=69 is used to guarantee stability according the CFL constraint. This plot shows that there are signi®cant trailing oscillations for both the

Galerkin and quadratic Petrov±Galerkin methods and to a lesser extent for the cubic Petrov±Galerkin method. All methods in this group however have a mild downstream phase error. As we decrease our mesh sizes to try to match the performance of the two Runge±Kutta characteristic schemes, we see in Fig. 3(a) and (c) that the trailing os-cillations and numerical di€usion are less pronounced (for all but the cubic Petrov±Galerkin method), but still persist for all three methods until we decrease_Dxto 1=180 and_Dtto 1=500. In this case the numerical solutions are comparable to that of the BRKC and FRKC methods with Dxˆ1=60 and Dtˆ1=10, however, the CPU re-quirement for these methods is two orders of magnitude larger than that required to achieve similar results using the BRKC or FRKC methods (see Table 6).

Fig. 4 presents the corresponding results for the streamline di€usion ®nite element method. This method requires the use of a control parameter C, and we present in Fig. 4 the plots for three values of this pa-rameter in each of the subplots (a)±(c). In Fig. 4(a) we

Table 7. Results for SDM withC_ˆ1, 0.1, and 0.0001, (CPU is in seconds)

C Dt Dx L2 Error L1Error CPU Figures

1.0 1/60 1/60 1:12331110ÿ1 _6:024287

10ÿ2 _{68.7 4(a)}

1/120 1/60 8:82436610ÿ2 _4:589660

10ÿ2 _{134.3 ±}

1/180 1/60 7:94420610ÿ2 _4:08759710ÿ2 _{197.9 ±}

1/500 1/60 6:78912310ÿ2 _3:44797710ÿ2 _{546.2 ±}

1/60 1/120 8:79549210ÿ2 _4:61388310ÿ2 _{141.7 ±}

1/120 1/120 5:11234010ÿ2 _2:56126910ÿ2 _{269.1 4(b)}

1/180 1/120 3:81253010ÿ2 _1:86724210ÿ2 _{402.1 ±}

1/500 1/120 2:34189410ÿ2 _1:10074010ÿ2 _{1080.9 ±}

1/60 1/180 7:93280310ÿ2 _4:12863410ÿ2 _{222.0 ±}

1/120 1/180 3:85246610ÿ2 _1:89132410ÿ2 _{406.8 ±}

1/180 1/180 2:47619210ÿ2 _1:17390110ÿ2 _{596.7 4(c)}

1/500 1/180 1:10254810ÿ2 _4:89809710ÿ3 _{1613.5 ±}

0.1 1/60 1/60 5:24131010ÿ2 _2:72392710ÿ2 _{68.6 4(a)}

1/120 1/60 3:18459810ÿ2 _1:58170410ÿ2 _{133.6 ±}

1/180 1/60 2:55461910ÿ2 _1:23893210ÿ2 _{198.2 ±}

1/500 1/60 1:85371610ÿ2 _8:62311110ÿ3 _{545.8 ±}

1/60 1/120 3:68363310ÿ2 _1:83877410ÿ2 _{140.4 ±}

1/120 1/120 1:51492410ÿ2 _6:961759

10ÿ3 _{270.4 4(b)}

1/180 1/120 9:51248710ÿ3 _4:181123

10ÿ3 _{402.6 ±}

1/500 1/120 4:54231510ÿ3 _1:807909

10ÿ3 _{1083.5 ±}

1/60 1/180 3:18060810ÿ2 _1:56310210ÿ2 _{218.5 ±}

1/120 1/180 1:06763610ÿ2 _4:75547610ÿ3 _{411.9 ±}

1/180 1/180 5:76052710ÿ3 _2:41195010ÿ3 _{601.4 4(c)}

1/500 1/180 2:25640210ÿ3 _8:11806710ÿ4 _{1623.3 ±}

0.0001 1/60 1/60 3:27975610ÿ2 _1:71554410ÿ2 _{69.1 4(a)}

1/120 1/60 1:62375610ÿ2 _8:35026510ÿ3 _{140.4 ±}

1/180 1/60 1:15685310ÿ2 _5:91394910ÿ3 _{201.2 ±}

1/500 1/60 6:68705510ÿ3 _3:31804510ÿ3 _{566.3 ±}

1/60 1/120 2:31049410ÿ2 _1:10831210ÿ2 _{149.7 ±}

1/120 1/120 7:79757510ÿ3 _3:34992010ÿ3 _{284.2 4(b)}

1/180 1/120 4:38092910ÿ3 _1:73133610ÿ3 _{412.4 ±}

1/500 1/120 2:05040010ÿ3 _7:43704210ÿ4 _{1113.4 ±}

1/60 1/180 1:99256710ÿ2 _9:38115110ÿ3 _{230.5 ±}

1/120 1/180 5:69755810ÿ3 _2:369440

10ÿ3 _{433.3 ±}

1/180 1/180 2:96499610ÿ3 _1:105808

10ÿ3 _{624.9 4(c)}

1/500 1/180 1:63955110ÿ3 _7:244163